Zero — Invented or Discovered?

It seems almost a bizarre question. Who thinks about whether zero was invented or discovered? And why is it important?

Answering this question, however, can tell you a lot about yourself and how you see the world.

Let’s break it down.

“Invented” implies that humans created the zero and that without us, the zero and its properties would cease to exist.

“Discovered” means that although the symbol is a human creation, what it represents would exist independently of any human ability to label it.

So do you think of the zero as a purely mathematical function, and by extension think of all math as a human construct like, say, cheese or self-driving cars? Or is math, and the zero, a symbolic language that describes the world, the content of which exists completely independently of our descriptions?

The concept is so basic it is routinely mastered by the pre-kindergarten set. Consider the equation 3-3=0. Nothing complicated about that. It is second nature to us that we can represent “nothing” with a symbol. It makes perfect sense now, in 2017, and it's so common that we forget that zero was a relatively late addition to the number scale.

Here's a fact that's amazing to most people: the zero is actually younger than mathematics. Pythagoras’s famous conclusion — that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides — was achieved without a zero. As was Euclid’s entire Elements.

How could this be? It seems surreal, given the importance the zero now has to mathematics, computing, language, and life. How could someone figure out the complex geometry of triangles, yet not realize that nothing was also a number?

Tobias Dantzig, in Number: The Language of Science, offers this as a possible explanation: “The concrete mind of the ancient Greeks could not conceive the void as a number, let alone endow the void with a symbol.” This gives us a good direction for finding the answer to the original question because it hints that you must first understand the concept of the void before you can name it. You need to see that nothingness still takes up space.

It was thought, and sometimes still is, that the number zero was invented in the pursuit of ancient commerce. Something was needed as a placeholder; otherwise, 65 would be indistinguishable from 605 or 6050. The zero represents “no units” of the particular place that it holds. So for that last number, we have six thousands, no hundreds, five tens, and no singles.

A happy accident of no great original insight, zero then made its way around the world. In addition to being convenient for keeping track of how many bags of grain you were owed, or how many soldiers were in your army, it turned our number scale into an extremely efficient decimal system. More so than any numbering system that preceded it (and there were many), the zero transformed the power of our other numerals, propelling mathematics into fantastic equations that can explain our world and fuel incredible scientific and technological advances.

But there is, if you look closely, a missing link in this story.

What changed in humanity that made us comfortable with confronting the void and giving it a symbol? And is it reasonable to imagine creating the number without understanding what it represented? Given its properties, can we really think that it started as a placeholder? Or did it contain within it, right from the beginning, the notion of defining the void, of giving it space?

In Finding Zero, Amir Aczel offers some insight. Basically, he claims that the people who discovered the zero must have had an appreciation of the emptiness that it represented. They were labeling a concept with which they were already familiar.

He rediscovered the oldest known zero, on a stone tablet dating from 683 CE in what is now Cambodia.

On his quest to find this zero, Aczel realized that it was far more natural for the zero to first appear in the Far East, rather than in Western or Arab cultures, due to the philosophical and religious understandings prevalent in the region.

Western society was, and still is in many ways, a binary culture. Good and evil. Mind and body. You’re either with us or against us. A patriot or a terrorist. Many of us naturally try to fit our world into these binary understandings. If something is “A,” then it cannot be “not A.” The very definition of “A” is that it is not “not A.” Something cannot be both.

Aczel writes that this duality is not at all reflected in much Eastern thought. He describes the catuskoti, found in early Buddhist logic, that presents four possibilities, instead of two, for any state: that something is, is not, is both, or is neither.

At first, a typical Western mind might rebel against this kind of logic. My father is either bald or not bald. He cannot be both and he cannot be neither, so what is the use of these two other almost nonsensical options?

A closer examination of our language, though, reveals that the expression of the non-binary is understood, and therefore perhaps more relevant than we think. Take, for example, “you’re either with us or against us.” Is it possible to say “I’m both with you and against you”? Yes. It could mean that you are for the principles but against the tactics. Or that you are supportive in contrast to your values. And to say “I’m neither with you nor against you” could mean that you aren’t supportive of the tactic in question, but won’t do anything to stop it. Or that you just don’t care.

Feelings, in particular, are a realm where the binary is often insufficient. Watching my children, I know that it's possible to be both happy and sad, a traditional binary, at the same time. And the zero itself defies binary categorization. It is something and nothing simultaneously.

Aczel reflects on a conversation he had with a Buddhist monk. “Everything is not everything — there is always something that lies outside of what you may think covers all creation. It could be a thought, or a kind of void, or a divine aspect. Nothing contains everything inside it.”

He goes on to conclude that “Here was the intellectual source of the number zero. It came from Buddhist meditation. Only this deep introspection could equate absolute nothingness with a number that had not existed until the emergence of this idea.”

Which is to say, certain properties of the zero likely were understood conceptually before the symbol came about — nothingness was a thing that could be represented. This idea fits with how we treat the zero today; it may represent nothing, but that nothing still has properties. And investigating those properties demonstrates that there is power in the void — it has something to teach us about how our universe operates.

Further contemplation might illuminate that the zero has something to teach us about existence as well. If we accept zero, the symbol, as being discovered as part of our realization about the existence of nothingness, then trying to understand the zero can teach us a lot about moving beyond the binary of alive/not alive to explore other ways of conceptualizing what it means to be.